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The given rational expression simplifies to (n^2 - 6)/(n^2 - 2). However, n cannot be equal to ±√2 and ±√5. These are the values that would make the denominator of the original or simplified expression equal to 0.

To simplify the given rational expression we must first factor both the numerator and the denominator. The expression is n^4 - 11n^2 + 30/ n^4 - 7n^2 + 10. We recognize this is a quadratic in terms of n^2. Hence, the numerator factors into (n^2 - 6)(n^2 - 5), and the denominator factors into (n^2 - 5)(n^2 - 2).

The simplified version of the expression is (n^2 - 6)/(n^2 - 2) but we need to take into account the restrictions for n which are that n cannot be equal to ±√2 and ±√5. These are the values that would make the denominator of the original or simplified expression equal to 0, thus undefined.

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The rational expression [tex](n^4 - 11n^2 + 30)/(n^4 - 7n^2 + 10)[/tex] simplifies to [tex](n^2 - 6)/(n^2 - 2)[/tex], with restrictions on n being ±√5 and ±√2 since these values would cause the original denominator to be zero.

To simplify the rational expression [tex]n^4 - 11n^2 + 30[/tex] divided by [tex]n^4 - 7n^2 + 10[/tex], we start by factoring both the numerator and the denominator.

Factor the numerator:

→ [tex]n^4 - 11n^2 + 30 = (n^2 - 5)(n^2 - 6)[/tex]

Factor the denominator:

→ [tex]n^4 - 7n^2 + 10 = (n^2 - 5)(n^2 - 2)[/tex]

After factoring, we can eliminate common terms.

The term ([tex]n^2 - 5[/tex]) is present in both the numerator and the denominator and can be cancelled out.

The simplified expression is [tex](n^2 - 6)/(n^2 - 2)[/tex].

We also need to state restrictions on the variable n. These restrictions come from the values that make any denominator zero, which are not allowed.

For the initial fraction, the restrictions are the values of n that make [tex]n^2 - 5[/tex] or [tex]n^2 - 2[/tex] equal to 0. Thus, n cannot be ±√5 or ±√2.

Final answer:

The perimeter of the polygon, given the sides (2x + 1), (4x – 10), (x + 1), and (12x + 25), is calculated by summing these expressions to get 19x + 17.

Explanation:

To find the perimeter of a four-sided polygon, we add together the lengths of all the sides. Given the sides (2x + 1), (4x – 10), (x + 1), and (12x + 25), the perimeter P in terms of x is calculated by summing these expressions:

P = (2x + 1) + (4x – 10) + (x + 1) + (12x + 25)

Now, we combine like terms:

P = 2x + 4x + x + 12x + 1 – 10 + 1 + 25

P = (2x + 4x + x + 12x) + (1 – 10 + 1 + 25)

P = 19x + 17

Therefore, the perimeter of the polygon in terms of x is 19x + 17.

The 15 days it takes for 8 goats and 8 cows to eat 17 acres of grass if it takes 2 goats and 3 cows 6 days to eat 2 acres of grass. it takes 6 goats and 5 cows 10 days to eat 8 acres of grass.

Fraction number consists of two parts one is the top of the fraction number which is called the numerator and the second is the bottom of the fraction number which is called the denominator.

We have:

it takes 2 goats and 3 cows 6 days to eat 2 acres of grass. it takes 6 goats and 5 cows 10 days to eat 8 acres of grass.

Number of days to eat 2 acres for  2 goats and 3 cows = 2/6

Number of days to eat 8 acres for 6 goats and 5 cows = 8/10

Now adding the fraction number of 2/6 and 8/10

[tex]\rm = \frac{2}{6}+ \frac{8}{10}[/tex]

= 68/60

= 15/17

Now the number of days is 15 days for the 17 acres.

Thus, the 15 days it takes for 8 goats and 8 cows to eat 17 acres of grass if it takes 2 goats and 3 cows 6 days to eat 2 acres of grass. it takes 6 goats and 5 cows 10 days to eat 8 acres of grass.

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We have been given the expression [tex] (a^8)^4 [/tex].

To solve this problem we will use one of the rules of the exponents called the Power Rule.

The power rule states that:

[tex] (x^m)^n=x^{m\times n}=x^{mn} [/tex]

In our case, x=a and the powers involved are m=8 and n=4.

Therefore, applying the power rule to the case given in our question we get:

[tex] (a^8)^4=a^{8\times 4}=a^{32} [/tex].

Therefore, Option D which says [tex] \boldsymbol{a^{32}} [/tex] is the correct option.

Answer:

C

Step-by-step explanation:

the interquartile range formula is the first quartile subtracted from the third quartile:

The question about how many pets two groups of students have cannot be answered without the correct data. Typically, such a question would involve computing descriptive statistics or organizing the data into a graph or table for easier comparison and interpretation.

Based on the information given in the prompt, which seems to contain a mistake as it does not directly correlate to the question about the two groups of students and their number of pets, we can not effectively conclude which of the following is true regarding the data about Group A and Group B without the appropriate data or statements to compare. Therefore, we need the correct data to proceed with the analysis. Generally, in math problems like this, we would compute the mean, median, mode, or range of the data to compare the two sets, or we could use more advanced statistics if required.

In a correct scenario, you could group the data differently by organizing it into a frequency table or creating a graph to visualize the distribution. Depending on the data shape, you might group it by intervals or categories. Grouping data helps to understand and interpret the data more easily. For instance, it is often easier to see trends and patterns in a histogram or bar chart than in a raw list of numbers.

To solve the equation (6x2n)÷8=15 for n, you must perform algebraic operations to isolate n. This involves multiplying by 8, then dividing by 6x, and finally by 2, to arrive at n = 20 / x.

To solve the equation (6x2n)÷8=15 for n, we need to follow several steps. This involves manipulating the equation using algebraic operations until we isolate the variable n on one side. Here is the step-by-step process:

If x is known, you can substitute the value of x into the equation to find the value of n. If x is not given, then this is the simplified form of the equation in terms of n and x.

A monthly of $60 can be spent for food after all the expenses and savings

are sorted.

A numerical expression is a mathematical statement written in the form of numbers and unknown variables. We can form numerical expressions from statements.

The numerical form of the given information can be formed as,

$(750 - 20 - 75 - 85 - 260 - 25).

= $60.

So, With an income of $750 and all the expenses including saving for for college worth of $250 the amount that can be spent on food is $60.

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Answer:

  30.0 cm

Step-by-step explanation:

You want the side length of an equilateral triangle with a height of 26 cm.

The altitude of the triangle divides it into two congruent 30°-60°-90° right triangles. The longer leg is the height: 26 cm. The hypotenuse is the longest of the three sides, which have the ratios ...

  1 : √3 : 2

That is, the side of the equilateral triangle is 2/√3 times the altitude.

  [tex]s = \dfrac{2}{\sqrt{3}}h = \dfrac{2(26\text{ cm})}{\sqrt{3}}\approx\boxed{30.0\text{ cm}}[/tex]

The length of each side is about 30.0 cm.

__

Additional comment

The side length is an irrational number of cm, approximately 30.022213997.... It rounds to 30.

Answer:

y = –3x + 4 and y = –3x – 4              no solution exists

y = –3x + 4 and 3y = –9x + 12           y=4-3x

y = –3x + 4 and y = -1/3x + 4              x=0, y=4

y = –3x + 4 and y = –6x + 8               x=4/3, y=0

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Step-by-step explanation:

The sides 6 mm, 8 mm and 10 mm forms a right angles triangle.

We have,

The sides of the triangle are of lengths 6 mm, 8 mm and  10 mm.

The longest side length is 10 mm.

Now, the sum of squares of the smaller sides is given as:

6² + 8² = 36 + 64 = 100 = 10²

As, the sum of squares of the smaller sides is equal to the square of the longer side.

So, by the Pythagorean Theorem, the sides form a right triangle.

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Answer:

No, because it doesn't satisfy the equation of the circumference

Step-by-step explanation:

A circle is the locus of points on the plane that are equidistant from a fixed point called the center.  For a circle whose center is the point

[tex]C=(a,b)[/tex]

and its radius is [tex]r[/tex], the ordinary equation of this circle is given by:

[tex](x-a)^2+(y-b)^2=r^2[/tex]

Since the circle is centered at the origin:

[tex]C=(a,b)=(0,0)\\\\Hence\\\\(x-0)^2+(y-0)^2=r^2\\\\x^2+y^2=r^2[/tex]

Now, let's find [tex]r[/tex] using the data provided. Evaluating the point (0,-4) into the equation:

[tex](0)^2+(-4)^2=r^2\\\\16=r^2\\\\r=\pm 4[/tex]

Thus the equation for the circle given by the problem is:

[tex]x^2+y^2=16[/tex]

In order to corroborate if the the point (2 3, 2) lie on the circle, we need to evaluate it into the equation and check if it satisfy the equation:

Note: I don't know what you mean with 2 3, so I will assume 3 cases:

[tex]2\hspace{3} 3=23\\2\hspace{3} 3=2*3=6\\2\hspace{3} 3=\frac{2}{3}[/tex]

First case:

[tex](23)^2+(2)^2=16\\\\533\neq16[/tex]

It doesn't satisfy the equation, therefore doesn't lie on the circle.

Second case:

[tex](6)^2+(2)^2=16\\\\40\neq16[/tex]

It doesn't satisfy the equation, therefore doesn't lie on the circle.

Third case:

[tex](\frac{2}{3} )^2+(2)^2=16\\\\\frac{40}{9} \neq16[/tex]

It doesn't satisfy the equation, therefore doesn't lie on the circle.

Final answer:

The point (3, 2) does not lie on the circle centered at the origin with a radius of 4 units because when its coordinates are plugged into the circle's equation x² + y² = 16, the result is not equal to the radius squared.

Explanation:

To determine if the point (3, 2) lies on the circle centered at the origin that contains the point (0, -4), we need to see if it satisfies the equation of the circle.

We know the radius of the circle is the distance from the center to the point (0, -4), which is 4 units since all points on a circle are equidistant from the center.

The general equation for a circle centered at the origin (0,0) is x² + y² = r², where r is the radius.

Our circle's equation is x² + y² = 16.

We plug in the coordinates of the point (3, 2) to see if it lies on the circle: 3² + 2² = 9 + 4 = 13, which is not equal to 16. Therefore, the point (3, 2) does not lie on the given circle.

Answer:

She only had $18 left

Step-by-step explanation:

To solve the expression 12 + 3 • 8, first perform the multiplication to get 24, then add this to 12 to get a final result of 36.

To evaluate the expression 12 + 3 • 8, you must follow the order of operations, often remembered by the acronym PEMDAS which stands for Parentheses, Exponents, Multiplication and Division, Addition and Subtraction, from left to right. As per PEMDAS, multiplication should be carried out before addition.

First, perform the multiplication: 3 times 8 is 24. Then, add the result to 12.

So, 12 + 24 equals 36. Therefore, the value of the expression 12 + 3 • 8 is 36.

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Answer:

4(base*height)/2

Answer:

The tens digit number is 9 and the ones digit number is 3.

Answer:

The arc length is [tex]32.84\ cm[/tex]    

Step-by-step explanation:

we know that

The circumference of a circle is equal to

[tex]C=2\pi r[/tex]

we have

[tex]r=36\frac{3}{5}\ cm=\frac{36*5+3}{5}=\frac{183}{5}\ cm[/tex]

substitute

[tex]C=2(3.14)(\frac{183}{5})=229.848\ cm[/tex]

Remember that

[tex]2\pi[/tex] radians subtends the complete circle of length [tex]229.848\ cm[/tex]

so

by proportion

Find the arc length by a central angle of [tex]2\pi/7[/tex] radians

[tex]\frac{229.848}{2\pi}=\frac{x}{2\pi/7}\\ \\x=229.848*(2\pi/7)/(2\pi)\\ \\x=`32.84\ cm[/tex]

The arc length is approximately 32.81 cm.

Step 1

To find the arc length intercepted by a central angle, we can use the formula:

[tex]\[ \text{Arc Length} = \text{radius} \times \text{central angle} \][/tex]

Given:

- Radius [tex](\( r \)[/tex]) = 36 3/5 cm

- Central angle [tex](\( \theta \)) = \( \frac{2\pi}{7} \)[/tex] radians

First, let's convert the radius to a decimal:

[tex]\[ \text{Radius} = 36 \frac{3}{5} \text{ cm} = 36.6 \text{ cm} \][/tex]

Step 2

Now, we can use the formula to find the arc length:

[tex]\[ \text{Arc Length} = 36.6 \times \frac{2\pi}{7} \]\[ \text{Arc Length} = 36.6 \times \frac{2 \times 3.14}{7} \]\[ \text{Arc Length} = 36.6 \times \frac{6.28}{7} \]\[ \text{Arc Length} = 36.6 \times 0.897 \]\[ \text{Arc Length} \approx 32.808 \][/tex]

Rounded to the nearest hundredth, the arc length is approximately 32.81 cm.

Answer:

C. Linearly, because the table shows that temperature increased by the same amount each month

Step-by-step explanation: